a) If $r$ is the radius of a circle then:

• Perimeter $= 2 \pi r$
• Area $= \pi r^2$  or  Area $=$ $\frac{1}{4}$ $\pi d^2$
• Diameter $(d) = 2r$

b) If $r$ is the radius of a circle then:

• Perimeter of a semi circle $= \pi r + 2r = (\pi + 2)r$
• Area of semi circle $=$ $\frac{1}{2}$ $\pi r^2$

c) If $r$ is the radius of a circle then:

• Perimeter of the quadrant $=$ $\frac{1}{4}$ $\times 2 \pi r + 2 r = ($ $\frac{\pi}{2}$ $+ 2)r$
• Area of the quadrant $=$ $\frac{1}{4}$ $\pi r^2$

d) If $R$ and $r$ are radii of two concentric circles, then

• Area enclosed between the two circle $= \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi (R+r)(R-r)$

e) If  $\triangle ABC$ is an equilateral triangle of side $a$ and height $h$, then $h =$ $\frac{\sqrt{3}}{2}$ $a$.

Also if $R$ and $r$ are the radii of the circumscribed and inscribed circles of $\triangle ABC$, then $R =$ $\frac{2h}{3}$ and $r=$ $\frac{h}{3}$

• Circumference of the circumcircle of  $\triangle ABC = 2\pi R = = 2 \pi \times$ $\frac{2h}{3}$ $=$ $\frac{4}{3}$ $\pi h$
• Area of the circumcircle $= \pi R^2 = \pi ($ $\frac{2h}{3}$ $)^2 =$ $\frac{4}{9}$ $\pi h^2$
• Circumference of the inscribed circle $= 2 \pi r =$ $\frac{2 \pi}{3}$ $h$
• Area of the inscribed circle $= \pi r^2 =$ $\frac{1}{9}$ $\pi h^2$

f) Important notes:

• If two circles touch internally, then the distance between their centers is equal to the difference of their radii.
• If tow circles touch externally. Then the distance between their centers is equal to the sum of their radii.
• The distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
• The number of revolutions completed by a rotating wheel in one minute $= \frac{Distance \ moved \ in \ one \ minute}{circumference}$